The Binary Home Range

In my foregoing post I criticized the Burt legacy (Burt 1943) for hampering progress in analysis of animal space use on a local scale. Analyses of the spatial pattern of GPS fixes – when this has been explicitly explored by multi-scale methods – consistently confirm a statistical fractal with dimension 0.9<D<1.2 rather than a paradigm-confirming to-dimensional area demarcation (albeit with fuzzy borders; see below). In an ideal world the D≈1 result should lead to hefty follow-up tests from the community of animal ecologists for the sake of verifying or falsifying the “home range as a fractal”-model and its behavioural-ecological implications. After all, the home range concept is a cornerstone of animal ecology. Nope. The Burt legacy still appears impenetrable. However, things finally seem to start rolling.

After a quarter of a century long invitation period  following the initial papers on the topic (Loehle 1990; Gautestad and Mysterud 1993, 1994) the application of fractal based approach that has spun off from the home range ghost concept (Gautestad and Mysterud 1995) has begun in small steps (Morellet et al. 2013; Campos et al. 2014). Also from the theoretical angle a development is finally surfacing along biophysically fascinating paths (Song et al. 2010; Boyer et al. 2012; Boyer and Romo-Cruz 2014; Boyer and Solis-Salas 2014; Boyer and Pineda 2016).

The D≈1 pattern, which represents “fix aggregation within fix aggregation within…”, needs to be understood in a multi-scaled context of animal space use, which is in conflict with basic assumptions under the standard home range concept and statistical methods. How should a fractal analysis proceed? For example, starting with a set of GPS fixes and analyzing it in a two-dimensional histogram, let the zero-count columns represent lack of incidence of GPS fixes (binary zeros), as opposed to non-zero columns (binary ones). The sum of the latter represents the home range representation, called incidence (I), at the actual grid resolution. The similar sums at other resolutions give a set of finer and coarser-scale representations of this “binary home range”. The initial analysis is simple:

  • The box counting method (Feder 1988). By studying the incidence function F(k) for all N fixes at linear resolutions k=1, 1/2, 1/4, 1/8, …, where k2=1 is the unit square that embeds all fixes, one gets the first result: is the pattern fractal-compliant or not? A power law-compliant regression; i.e., the regression line for log[F(k)] over a range of log(k) is approximately linear, will satisfy this criterion. As a rule-of-thumb, such linearity (power law compliance, called “self-similar”) should persist over a relative scale range of 100 or larger. Thus, N should be large to allow for this test. Thus, be aware of the dilution effect and the space fill effect which both tend to artificially narrow the scale-free range (Gautestad and Mysterud 2012).
  • The fractal dimension. The negative of the slope of log[F(k)] over a range of log(k) gives an estimate of D. For an area-confirming kind of home range; i.e., an object satisfying D=2, one should expect number of non-empty squares to increase proportionally with 1/k2. If the behaviour is compliant with a space-constrained random walk of the classical kind (a common modelling approach) and thus leading to quite fuzzy area demarcations, one should expect 1.4<D<2. See examples in Chapter 3 of my book.  If D≈1; i.e, incidence increases proportionally with 1/k1 = 1/k, the paradigmatic HR concept has a problem. Under this condition, the pattern shows core areas within core areas within …, even in a hypothetically homogeneous environment (Gautestad and Mysterud 2005; see also this post).
  • The home range ghost. As an alternative analysis to F(k), study incidence at a given resolution (k<1) as a function of sample size of fixes N=1, 2, 4, 8, 16, … Nmax. In other words, study F(N) at a given k rather than F(k) at a given N. Then repeat the procedure for different scales k, as explained above. If the given individual’s space use is both scale-free (spatially self-similar) and also influenced by site fidelity (resulting in a home range-emerging kind of movement) you should find a “balancing” resolution k* where log-transformed incidence log(I) expands approximately linear with log(N). The basic home range ghost model predicts ∝ N0.5; i.e., at this scale the binary home range area expands proportionally with the square root of number of fixes in the sample. This “paradoxical” N-dependency persists after factors like serial auto-correlation in the fixes and drifting space use has been accounted for. It is an emergent property from an animal that is utilizing its home range in a scale-free manner. The regression line’s intercept with N=1; i.e., at log(N)=0, then gives “the characteristic scale of space use” (CSSU), which is this theory’s replacement of the traditional home range area concept. By the way, the CSSU can also be inferred in a complementary manner from the box counting method for estimating D, as explained in this post.

While the theory for treating a home range as an area predicts an area asymptote for I(N); i.e., in compliance with the Burt model, the home range ghost model predicts a power law model. While the former in an ecological context invites to demarcate and study a home range area, the alternative model requires application of k* as an ecological proxy variable (larger k* resembles the idea of a larger home range).

  • In the classical framework, an area demarcation makes sense, since it represents a given percentage of a two-dimensional stationary statistical distribution of space use.
  • In the binary home range framework (based on the parallel processing-compliant space use) the home range area is not intrinsically stationary since it is N-dependent (lacking an area asymptote). Instead, k* represents the stationary parameter.
  • Since k* represents a linear scale, squaring it makes (k*)2 proportional with the parameter c in I=cNz (where the power exponent z≈0.5 in the ideal model).
  • As shown in Eq. 17 in my book, he relationship between D from F(k) and k* from F(N)=I(N) is reflected in the formula I(N) = cN(1-D/2). See also Gautestad and Mysterud (2010).

At first sight a binary representation of a home range may appear to imply information loss in comparison to a continuous variable like a utilization distribution (“density surface”). However, D, k*, c and z in the binary modelling approach are all continuous quantities. Further, this framework’s analogue to the utilization distribution is a surface where the local variation of intra-home range density of fixes is replaced by intra-home range variation of (1/k*)2 = 1/c. All these novel system descriptors provide the alternative toolbox for ecological research on animal space use.

In this example (simulated data) a large set of “fixes” is collected from a scenario where intra-home range intensity of space use varies between the four quadrats at k=1/2 relative to total area k=1. The parameter c is then estimated from subsets of fixes in respective quadrats. Details will become available upon publication (Gautestad in prep.).

The latter analysis – local variation in 1/c – implies splitting the spatial GPS scatter of fixes into spatial sub-sections (e.g., simple squares, or polygon demarcations of local habitat patches based on some ecological criteria). The embedded subset of fixes within each subsection of the home range is then subject of fractal analysis for the respective intra-home range locations. A preliminary pilot test of the latter involving empirical data rather than simulations was presented in this post.

As  shown by the references above, applications and theoretical developments of the fractal home range model has only recently commenced beyond our own series of papers. Why is the D≈1 property of animal space use so difficult to relate to in the community of wildlife ecologists? It can’t be due to lack of proposals for concrete methodological guidelines (Gautestad and Mysterud 2005, 2012; Gautestad 2012,  Gautestad and Mysterud 2013; Gautestad et al. 2013). Further, it shouldn’t be due to lack of theory-supporting empirical results (Gautestad and Mysterud 1993, 1995; Gautestad et al.1998, 2013). The basic statistical methods to test or apply the theory are also quite straight-forward, as summarized above. In particular, the key method to translate the spatial scatter of GPS fixes as a binary presence/absence of fixes in respective virtual grid cells and repeating this analysis over a range of grid resolutions is easy to implement both in R and other statistical packages.

Optimistically, I’m still waiting for the ketchup effect with respect to frenetic research on the home range as a fractal; the binary and multi-scaled representation of space use. This requires a critical evaluation of the Burt legacy. Thus, I may still have to wait for a while…


Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Boyer, D., and J. C. R. Romo-Cruz. 2014. Solvable random walk model with memory and its relations with Markovian models of anomalous diffusion. Physical review E. 90:1-12.

Boyer, D., and C. Solis-Salas. 2014. Random walks with preferential relocations to places visited in the past and their application to biology. arXiv 1403.6069v1:1-5.

Boyer, D., and I. Pineda. 2016. Slow Lévy flights. arXiv:1509.01315v2

Burt, W.H., 1943. Territoriality and home range concepts as applied to mammals. J. Mammal. 24:346-352.

Campos, F. A., M. L. Bergstrom, A. Childers, J. D. Hogan, K. M. Jack, A. D. Melin, K. N. Mosdossy et al. 2014. Drivers of home range characteristics across spatiotemporal scales in a Neotropical primate, Cebus capucinus. Animal Behaviour 91:93-109.

Feder, J. 1988, Fractals. New York, Plenum Press.

Gautestad, A. O., and I. Mysterud. 1993. Physical and biological mechanisms in animal movement processes. J. Appl. Ecol. 30:523-535.

Gautestad, A. O., and I. Mysterud. 1994. Are home ranges fractals? Landscape Ecol. 2:143-146.

Gautestad, A. O., and I. Mysterud. 1995. The home range ghost. Oikos 74:195-204.

Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.

Gautestad, A. O., and I. Mysterud. 2010. The home range fractal: from random walk to memory dependent space use. Ecological Complexity 7:458-470.

Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9:2332-2340.

Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.

Gautestad, A. O., and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

Gautestad, A. O., L. E. Loe, and A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.

Loehle, C. 1990. Home range: a fractal approach. Landscape Ecology 5:39-52.

Morellet, N., C. Bonenfant, L. Börger, F. Ossi, F. Cagnacci, M. Heurich, P. Kjellander et al. 2013. Seasonality, weather and climate effect home range size in roe deer across a wide latitudinal gradient within Europe. Journal of Animal Ecology 82:1326-1339.

Song, C., T. Koren, P. Wang, and A.-L. Barabási. 2010. Modelling the scaling properties of human mobility. Nature Physics 6:818-823.


Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis

The term “Home range” (HR) generally follows Burt’s (1943) definition, the area traversed by the individual in its normal activities of food gathering, mating, and caring for young. Occasional sallies outside of the area, perhaps exploratory in nature, should not be considered as home range. However, with respect to fine-grained perception of a HR, Burt’s definition seems to have guided – in fact cemented – the HR concept into a too narrow and partly misleading perception of individual space use. Hence, in my view the Burt definition is hampering progress in this important field of animal ecology.

From the perspective of a regional map, an individual’s home range is a zero-dimensional dot. When zooming in towards medium scale, it makes sense to demarcate a home range as a two-dimensional area (or a three-dimensional volume in the context of aquatic or marine systems). The challenge to define HR borders at this scale is reflected in Burt’s second part of his HR definition, leaving some “occasional sallies” outside the polygon, ellipse, isopleth-based demarcation, or whatever kind of area that is differentiating “inside” from “outside”. For more than 60-70 years Ecology has lived well with this medium-resolution definition of a HR. However, recently the Burt definition has become an obstacle as its application has been extended towards finer scales as a consequence of better data.

Modern methods of path collection have greatly improved the length of data series both with respect to extent and temporal resolution.  Further, sophisticated statistical analyses for HR demarcations (e.g., KDE and Brownian Bridge) have replaced polygons and other initial protocols for area demarcation. However, the improved quality and quantity of space use data has not made life easier for statistical analyses. Rather, the “inside” versus “outside” issue has surfaced as a core challenge for studies of animal space use and its ecological aspects. Introducing a more sophisticated HR concept by differentiating between the HR and its intra-HR “core areas” has created new challenges. As a result, the Burt definition’s shortcomings are now surfacing and causing seeds of confusion and controversy. In this post I claim that these issues go beyond the normal maturing of ecological methods, where more sophisticated approaches are evolving as a natural progression from (i.e., refinement of) the older approaches. Hence, the present headline, “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“.

The HR is a challenging concept per se, as illustrated by this mixed-species flock of gulls at the western coast of Norway (observe the relatively rare glaucous gull Larus hyperboreus, number 4 from the right). Inter- and intra-species HR overlap, temporal HR stability and other quantifications require statistically precise variables that are spun from a realistic statistical-mechanical framework. Photo: AOG.


Several assumptions are included in the Burt definition, explicitly or implicitly:

  1. HR as an area. Our non-standard approach to analyze the spatial scatter of GPS fixes across a range of spatial resolutions has shown that the respective individuals’ home range better satisfy the concept of a statistical fractal; i.e., a non-Euclidean fractional dimension typically in the range 0.9<D<1.2, than an Euclidean area (D=2, surrounded by some Euclidean dots with D=0 called occasional sallies). Thus, treating a HR as a two-dimensional object may be non-realistic and thus misleading for ecological research involving finer spatial scale than the medium “Burt scale” referred to above.
  2. HR as a stationary area. Recent theoretical work on the HR concept is still implicitly or explicitly assuming that any realistic HR model (for example, as reflected in the output from a simulation) should satisfy a stationary HR area. In statistical-mechanical terms, HR as an area requires a scale-specific kind of habitat utilization ( a Brownian motion-like model that is extended with HR-mimicking mechanics satisfies this criterion). In other words, after accounting for the trivial effects from serial autocorrelation (high-frequency path sampling), drifting HR, and small-sample statistical artifacts, a HR model that still shows a non-stationary range as a function of sample size of fixes should be considered pathological. Non-stationary area implies that the model does not represent HR behaviour. By clinging to this view, one has to disregard a growing set of empirical analyses spanning many species confirming sample-size-dependent HR area (“the home range ghost”). Such non-stationary space use – remaining after the trivial sample effects have been accounted for – becomes a statistical paradox from the perspective of the Burt paradigm, but a trivial property from the perspective of scale-free space use under influence of spatial memory. The latter leads to a set of GPS fixes satisfying a statistical fractal object rather than an integer-dimensional object.
  3. Occasional sallies versus intra-HR displacements. Turning from the Eulerian to the Lagrangian perspective of space use, the Burt definition forces upon us a distinction between “normal” HR activities and occasional long-distance moves spanning larger ranges than the HR. When successive inter-fix distances are presented in a histogram (frequency of “step lengths” as a function of length), the typical pattern shows the majority of such step lengths located in the small-step bins, some steps in medium-range bins and some (occasional) steps of very large extent. Thus, in the spirit of the Burt definition several attempts have been proposed to differentiate true HR movement from occasional sallies. However, the methodology remains problematic and quasi-objective. On the other hand, if both axes in the histogram are log-transformed, the regression line typically becomes approximately linear over a broad part of the range of step lengths. This is a strong indication of a scale-free space use process (frequent fine-scale moves intermingled with fewer coarse-scale moves, in a mathematically fractal-compliant manner). Thus, while occasional sallies represent a stone in the shoe under the paradigmic HR concept, the issue evaporates under the fractal HR concept.

To conclude, the HR concept as defined by Burt is a feasible and important part of animal ecology when applied on medium scales, but becomes misleading and counter-productive when stretched towards finer spatial analyses.

Interestingly, many of the key ecological variables that are studied under the Burt paradigm (local intensity of space use, habitat preference, HR size, HR overlap, core areas, HR stability, etc.) have their quantifiable counterparts under the emerging “HR as a fractal” model. Unfortunately, the Burt legacy is hampering a broader exploration and critical evaluation of the latter.


Burt, W.H., 1943. Territoriality and home range concepts as applied to mammals. J. Mammal. 24:346-352.

The Mysterious Taylor’s Power Law – Part IV

Time to verify theoretical coherence between scale-free population abundance and scale-free space use at the individual level! In this part IV of the Taylor law presentation I analyze a simulated set of GPS fixes rather than studying population abundance. In other words, how does the variance-mean relationship in local density of fixes from the multi-scaled random walk model (MRW) resemble V(M) in a local population under the Zoomer model condition? Through the history spanning more than 1,000 papers this acid test has never previously been successfully performed. My presentation also illustrates the so-called Z-paradox, and how it is resolved under the parallel processing framework for animal space use.

The illustration to the right shows the spatial “home range” scatter from a model individual complying with the MRW model. The number of fixes pr. grid cell of resolution k=1:128 (128×128 cells within the defined arena) shows the commonly observed multi-modal utilization distribution. In this case, and in compliance with all our empirical tests on real data, the distribution is self-similar (fractal), with fractal dimension D≈1.

As described in Chapter 7 of my book, V(M) analysis of fixes shows compliance with Taylor’s power law V=aMb with power exponent b≈2. The illustration above – with log-transformed data – shows “transects” of local fix abundance at various spatial resolutions (see Note below). For example, the red dots illustrate V(M) at resolution k=1:128 relative arena size k≡1, where each dot is a given V(M) pair in the “red” set of 128 transect rows. The blue dots show the similar set of transects at resolution 1:64, and so on towards coarser spatial scales 1:32, 1:16 and 1:8. The black dots with regression line show the over-all V(M) for all grid cells at respective resolutions, instead of splitting the cells into transects at respective scales. According to standard, “well-behaved” statistics (Poisson-compliant, where b=1), one should expect the black V(M) series to overlap the transect sets. In other words, the intercept log(a) should be similar when studying V(M) at a coarse resolution. In short, the variance of the total set equals the sum of the variance of the parts. Why this is not true here in the context of complex (scale-free) space use, is previously explained as the Z-paradox. This paradox emerges when the classical theory for variance is applied on non-classical space use. By applying the parallel processing theory the paradox is resolved, as shown below!

By multiplying the variance for each V(M) pair with a scale-dependent coefficient equal to the square root of 1/k (so-called rescaling), we get full compliance between mean and variance at respective scales when the V(M) sets from respective scales are superimposed. In other words, this rescaling is represented by the coefficient a in Taylor’s power law under the present model conditions where we study cross-scale scaling of parallel processing-compliant space use at the individual level. The magnitude of a in this context increases with the relative scale range when comparing V(M) at different resolutions. For example, for = 1/128 we have a = √128 = 11.31. Thus, log(a) = 1.05, which is very close to the y-axis intercept log(a) = 0.90 that was calculated for the log(V,M) regression at the chosen reference scale k≡1 in the pilot test above (black dots).

Why square root? Because b=2 implies that the standard deviation (SD), which equals the square root of V,  is proportional with the mean, M. Thus, the present statistical-mechanical framework implies that its V(M) statistics are compliant with the properties of a Cauchy distribution (Lévy stable distribution of index 1) rather than the normal Gaussian under which variance is proportional with the mean.

The illustrative result above comes from a single set of GPS fixes. If the procedure had been replicated, the average V(M) scatter from respective replicates would in the limit of large number of replicates show less scatter around the regression line.

From a statistical-mechanical perspective this result verifies coherence between time average and ensemble average of space use by a “particle” or a set of particles, but hereby – for the first time to my knowledge – from the context of a complex (scale-free) process with b>>1.  

GPS fixes from MRW represent the time evolution of a single particle’s space use (in statistical terms), and the ensemble average from the Zoomer model shows a population of particles’ space use at a given point in time (after some initial evolution of implicit individual movement).

For the first time a statistical-mechanical framework is able to show coherence between individual level and population level dynamics in the context of V(M) statistics, in compliance with Taylor’s power law and thus a broad range of empirical data. In technical terms, the dual-level MRW/Zoomer model simulates a so-called Lévy-stable process (Mandelbrot 1983) over a scale range. Hence, hereby the theoretical regime of statistical fractals is connected to Taylor’s power law through novel extensions of standard (Markov-compliant and scale-specific) statistical-mechanics!

In the next part of this series I will illustrate Taylor’s power law and its scale-free property using my own empirical data on a local population of aphids.


  • In compliance with Hanski (1982) I have adjusted for small-sample artifacts by modifying Taylor’s law to V = aMb + M. This modification was not performed in my book’s presentation of the same data (Figure 43).


Hanski, I. 1982. On patterns of temporal and spatial variation in animal populations. Ann. zool. Fermici 19: 21—37

Mandelbrot, B. B. 1983. The fractal geometry of nature. New York, W. H. Freeman and Company.